Ramsey Theorems for Knots, Links and Spatial Graphs

Abstract

An embedding $f:G \to {{\mathbf {R}}^3}$ of a graph $G$ into ${{\mathbf {R}}^3}$ is said to be linear if each edge $f(e)\quad (e \in E(G))$ is a straight line segment. It will be shown that for any knot or link type $k$, there is a finite number $R(k)$ such that every linear embedding of the complete graph ${K_n}$ with at least $R(k)$ vertices $(n \geqslant R(k))$ in ${{\mathbf {R}}^3}$ contains a knot or link equivalent to $k$.